M\"untz spaces and Remez inequalities

Two relatively long-standing conjectures concerning M\"untz polynomials are resolved. The central tool is a bounded Remez type inequality for non-dense M\"untz spaces.Comment: 5 page

A Markov-type inequality for the derivatives of constrained polynomials

AbstractMarkov's inequality asserts that max−1⩽x⩽1|p′(x)|⩽n2max−1⩽x⩽1|p(x)| (1) for every polynomial of degree at most n. The magnitude of supp∈Smax−1⩽x⩽1|p′(x)|max−1⩽x⩽1|p(x)| (2) was examined by several authors for certain subclasses S of Πn. In this paper we introduce S = Snm(r) (0 ⩽ m ⩽ n, 0 < r ⩽ 1), the set of those polynomials from Πn which have all but at most m zeros outside the circle with center 0 and radius r, and establish the exact order of the above expression up to a multiplicative constant depending only on m

Remez-type inequalities and their applications

AbstractThe Remez inequality gives a sharp uniform bound on [−1, 1] for real algebraic polynomials p of degree at most n if the Lebesgue measure of the subset of [−1, 1], where |;p|; is at most 1, is known. Remez-type inequalities give bounds for classes of functions on a line segment, on a curve or on a region of the complex plane, given that the modulus of the functions is bounded by 1 on some subset of prescribed measure. This paper offers a survey of the extensive recent research on Remez-type inequalities for polynomials, generalized nonnegative polynomials, exponentials of logarithmic potentials and Müntz polynomials. Remez-type inequalities play a central role in proving other important inequalities for the above classes. The paper illustrates the power of Remez-type inequalities by giving a number of applications

On the oscillations of the modulus of Rudin-Shapiro polynomials around the middle of their ranges

Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. The graphs of $R_k$ on the period suggest many zeros of $R_k(t)-n$ in a dense fashion on the period. Let $N(I,R_k-n)$ denote the number of zeros of $R_k-n$ in an interval $I := [\alpha,\beta] \subset [0,2\pi]$. Improving earlier results stated only for $I := [0,2\pi]$, in this paper we show that $$\frac{n|I|}{8\pi} - \frac{2}{\pi} (2n\log n)^{1/2} - 1 \leq N(I,R_k-n) \leq \frac{n|I|}{\pi} + \frac{8}{\pi}(2n\log n)^{1/2}\,,\qquad k \geq 2\,,$$ for every $I := [\alpha,\beta] \subset [0,2\pi]$, where $|I| = \beta-\alpha$ denotes the length of the interval $I$.Comment: 7 page

The uniform closure of non-dense rational spaces on the unit interval

AbstractLet Pn denote the set of all algebraic polynomials of degree at most n with real coefficients. Associated with a set of poles {a1,a2,…,an}⊂R⧹[-1,1] we define the rational function spaces Pn(a1,a2,…,an):=f:f(x)=b0+∑j=1nbjx-aj,b0,b1,…,bn∈R.Associated with a set of poles {a1,a2,…}⊂R⧹[-1,1], we define the rational function spacesP(a1,a2,…):=⋃n=1∞Pn(a1,a2,…,an).It is an interesting problem to characterize sets {a1,a2,…}⊂R⧹[-1,1] for which P(a1,a2,…) is not dense in C[-1,1], where C[-1,1] denotes the space of all continuous functions equipped with the uniform norm on [-1,1]. Akhieser showed that the density of P(a1,a2,…) is characterized by the divergence of the series ∑n=1∞an2-1.In this paper, we show that the so-called Clarkson–Erdős–Schwartz phenomenon occurs in the non-dense case. Namely, if P(a1,a2,…) is not dense in C[-1,1], then it is “very much not so”. More precisely, we prove the following result.TheoremLet {a1,a2,…}⊂R⧹[-1,1]. Suppose P(a1,a2,…) is not dense in C[-1,1], that is,∑n=1∞an2-1<∞.Then every function in the uniform closure of P(a1,a2,…) in C[-1,1] can be extended analytically throughout the set C⧹{-1,1,a1,a2,…}